Question

1. (5 points) convert to an exponential equation.

\\(\log_{4} 64 = 3\\)

1. (5 points) convert to a logarithmic equation.

\\(3^{3} = 27\\)

1. (10 points) evaluate using the change-of-base formula. round to three decimal places.

(a) \\(\log_{14} 15\\)
(b) \\(\log_{3} 22\\)

Explanation:

Step1: Convert logarithmic equation to exponential form

$$\log_b x = y \iff b^y = x$$

$$\log_4 64 = 3 \iff 4^3 = 64$$

Step2: Convert exponential equation to logarithmic form

$$b^y = x \iff \log_b x = y$$

$$3^3 = 27 \iff \log_3 27 = 3$$

Step3: Apply change-of-base formula for 5(a)

$$\log_{14} 15 = \frac{\ln 15}{\ln 14}$$

Step4: Calculate and round 5(a)

$$\log_{14} 15 \approx \frac{2.70805}{2.63906} \approx 1.026$$

Step5: Apply change-of-base formula for 5(b)

$$\log_3 22 = \frac{\ln 22}{\ln 3}$$

Step6: Calculate and round 5(b)

$$\log_3 22 \approx \frac{3.09104}{1.09861} \approx 2.814$$

Answer:

1. \(4^3 = 64\)
2. \(\log_3 27 = 3\)
3. (a) \(1.026\)

(b) \(2.814\)